Theorem mexval 35489

Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mexval.k	⊢ 𝐾 = (mTC‘𝑇)
mexval.e	⊢ 𝐸 = (mEx‘𝑇)
mexval.r	⊢ 𝑅 = (mREx‘𝑇)

Assertion

Ref	Expression
mexval	⊢ 𝐸 = (𝐾 × 𝑅)

Proof of Theorem mexval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mexval.e	. 2 ⊢ 𝐸 = (mEx‘𝑇)
2		fveq2 6858	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
3		mexval.k	. . . . . 6 ⊢ 𝐾 = (mTC‘𝑇)
4	2, 3	eqtr4di 2782	. . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
5		fveq2 6858	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
6		mexval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
7	5, 6	eqtr4di 2782	. . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
8	4, 7	xpeq12d 5669	. . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅))
9		df-mex 35474	. . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
10		fvex 6871	. . . . 5 ⊢ (mTC‘𝑡) ∈ V
11		fvex 6871	. . . . 5 ⊢ (mREx‘𝑡) ∈ V
12	10, 11	xpex 7729	. . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V
13	8, 9, 12	fvmpt3i 6973	. . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
14		xp0 6131	. . . . 5 ⊢ (𝐾 × ∅) = ∅
15	14	eqcomi 2738	. . . 4 ⊢ ∅ = (𝐾 × ∅)
16		fvprc 6850	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
17		fvprc 6850	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅)
18	6, 17	eqtrid 2776	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
19	18	xpeq2d 5668	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅))
20	15, 16, 19	3eqtr4a 2790	. . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
21	13, 20	pm2.61i 182	. 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅)
22	1, 21	eqtri 2752	1 ⊢ 𝐸 = (𝐾 × 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ∅c0 4296   × cxp 5636  ‘cfv 6511  mTCcmtc 35451  mRExcmrex 35453  mExcmex 35454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-mex 35474

This theorem is referenced by:  mexval2  35490  msubff  35517  msubco  35518  msubff1  35543  mvhf  35545  msubvrs  35547